Average Error: 31.1 → 0.0
Time: 1.9m
Precision: 64
Internal Precision: 128
\[\frac{x - \sin x}{x - \tan x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.06381524695988129 \lor \neg \left(x \le 0.06507269604379932\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Error

Bits error versus x

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Your Program's Arguments

Results

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Derivation

  1. Split input into 2 regimes

  2. if x < -0.06381524695988129 or 0.06507269604379932 < x

    1. Initial program 0.0

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Initial simplification0.0

      \[\leadsto \frac{x - \sin x}{x - \tan x}\]
    3. Using strategy rm

    4. Applied div-sub0.0

      \[\leadsto \color{blue}{\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}}\]
    5. Using strategy rm

    6. Applied add-log-exp0.0

      \[\leadsto \frac{x}{x - \tan x} - \color{blue}{\log \left(e^{\frac{\sin x}{x - \tan x}}\right)}\]

    if -0.06381524695988129 < x < 0.06507269604379932

    1. Initial program 62.8

      \[\frac{x - \sin x}{x - \tan x}\]

    2. Initial simplification62.8

      \[\leadsto \frac{x - \sin x}{x - \tan x}\]

    3. Taylor expanded around 0 0.0

      \[\leadsto \color{blue}{\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.06381524695988129 \lor \neg \left(x \le 0.06507269604379932\right):\\ \;\;\;\;\frac{x}{x - \tan x} - \log \left(e^{\frac{\sin x}{x - \tan x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{9}{40} \cdot {x}^{2} - \left(\frac{27}{2800} \cdot {x}^{4} + \frac{1}{2}\right)\\ \end{array}\]

Runtime

Time bar (total: 1.9m)Debug logProfile

herbie shell --seed 2018336 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  (/ (- x (sin x)) (- x (tan x))))